[seqfan] Re: Largest subsets of {1, ..., n} such that no difference is ...

[Neil Sloane]

a,b,c,d are now A100719, A131849, A362914, A362915.




On Mon, May 15, 2023 at 10:05 PM Rob Pratt via SeqFan <seqfan at list.seqfan.eu>
wrote:

> I confirmed your values for (c).
>
> (a): https://oeis.org/A100719
> (b): https://oeis.org/A131849
> (d): 1 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 6 6 6 6 6 6 6 6
> 6 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
> 8 8 8 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 12 12 12 12 12 12 12
>
> -----Original Message-----
> From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of Neil Sloane
> Sent: Monday, May 15, 2023 4:17 PM
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Largest subsets of {1, ..., n} such that no difference
> is ...
>
> EXTERNAL
>
> Dear Seqfans, A recent talk by Ben Green studies the density of, for
> example, the largest subset of {1..n} such that no difference is:
> (a) a square, (b) a prime - 1, (c) a prime, or (d) a prime + 1.
>
> So let's look at the actual sizes, not the density.
>
> I started off with (c), so let a(n) = max subset of {1..n} such that no
> difference is a prime.  For n=1..11 I get
> 1,2,2,2,2,2,2,2,3,3,4
> which seems not to be in the OEIS.
> The examples where a(n) increases are {1}, {1,2}, {1,5,9}, {1,2,10,11}.
> Could someone check?
>
> And what about (a), (b), and (d)?
>
> Best regards
> Neil
>
> Neil J. A. Sloane, Chairman, OEIS Foundation.
> Also Visiting Scientist, Math. Dept., Rutgers University,
> Email: njasloane at gmail.com
>
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